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To solve the problem of writing the expression \(x^2 - 3x + 5\) in the form \((x + a)^2 + b\) where \(a\) and \(b\) are improper fractions, we can complete the square:

1. Start with the given quadratic expression:
   \[
   x^2 - 3x + 5
   \]

2. To complete the square, focus on the quadratic and linear terms, \(x^2 - 3x\).

3. Find the term needed to complete the square:
   - Take half of the coefficient of \(x\) (which is \(-3\)), so \(\frac{-3}{2} = -\frac{3}{2}\).
   - Square this value: \(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\).

4. Rewrite the expression by adding and subtracting \(\frac{9}{4}\) inside the expression:
   \[
   x^2 - 3x + 5 = (x^2 - 3x + \frac{9}{4}) + 5 - \frac{9}{4}
   \]

5. Simplify inside the square:
   \[
   (x^2 - 3x + \frac{9}{4}) = \left(x - \frac{3}{2}\right)^2
   \]

6. Combine the constants:
   \[
   5 - \frac{9}{4} = \frac{20}{4} - \frac{9}{4} = \frac{11}{4}
   \]

7. So, the expression is:
   \[
   (x - \frac{3}{2})^2 + \frac{11}{4}
   \]

Therefore, the expression in the required form is:
\[
(x - \frac{3}{2})^2 + \frac{11}{4}
\]

If we follow the form given in the question \((x + a)^2 + b\), with \(a = -\frac{3}{2}\) and \(b = \frac{11}{4}\), both of which are improper fractions.

Would you like more details or have any other questions?

Here are some related questions:
1. How do you complete the square for different types of quadratic expressions?
2. Can completing the square be used to solve quadratic equations?
3. What are other methods to transform a quadratic expression into a different form?
4. How can the vertex form of a quadratic equation help in graphing parabolas?
5. What are the real-world applications of completing the square?

**Tip:** Completing the square is particularly useful when converting a quadratic expression into vertex form, which makes analyzing and graphing the function easier.

Write x^2 - 3x + 5 in the form (x + a)^2 + b where a and b are improper fractions.

Learn how to rewrite the quadratic expression x^2 - 3x + 5 in the form (x + a)^2 + b using the completing the square method. This step-by-step solution covers key algebraic concepts and formulas suitable for Grades 9-11.

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